This paper revisits classical works of Rauch (1963, et al. 1965) and develops a novel method for maximum likelihood (ML) smoothing estimation from incomplete information/data of stochastic state-space systems. Score function and conditional observed information matrices of incomplete data are introduced and their distributional identities are established. Using these identities, the ML smoother $\widehat{x}_{k\vert n}^s =\argmax_{x_k} \log f(x_k,\widehat{x}_{k+1\vert n}^s, y_{0:n}\vert\theta)$, $k\leq n-1$, is presented. The result shows that the ML smoother gives an estimate of state $x_k$ with more adherence of loglikehood having less standard errors than that of the ML state estimator $\widehat{x}_k=\argmax_{x_k} \log f(x_k,y_{0:k}\vert\theta)$, with $\widehat{x}_{n\vert n}^s=\widehat{x}_n$. Recursive estimation is given in terms of an EM-gradient-particle algorithm which extends the work of \cite{Lange} for ML smoothing estimation. The algorithm has an explicit iteration update which lacks in (\cite{Ramadan}) EM-algorithm for smoothing. A sequential Monte Carlo method is developed for valuation of the score function and observed information matrices. A recursive equation for the covariance matrix of estimation error is developed to calculate the standard errors. In the case of linear systems, the method shows that the Rauch-Tung-Striebel (RTS) smoother is a fully efficient smoothing state-estimator whose covariance matrix coincides with the Cram\'er-Rao lower bound, the inverse of expected information matrix. Furthermore, the RTS smoother coincides with the Kalman filter having less covariance matrix. Numerical studies are performed, confirming the accuracy of the main results.

翻译：本文重新审视了Rauch（1963年及1965年等人的经典工作），并提出了一种新方法，用于从随机状态空间系统的不完全信息/数据中进行最大似然平滑估计。本文引入了不完全数据的得分函数和条件观测信息矩阵，并建立了其分布恒等式。基于这些恒等式，提出了最大似然平滑器 $\widehat{x}_{k\vert n}^s =\argmax_{x_k} \log f(x_k,\widehat{x}_{k+1\vert n}^s, y_{0:n}\vert\theta)$，其中 $k\leq n-1$。结果表明，与最大似然状态估计器 $\widehat{x}_k=\argmax_{x_k} \log f(x_k,y_{0:k}\vert\theta)$（其中 $\widehat{x}_{n\vert n}^s=\widehat{x}_n$）相比，最大似然平滑器对状态 $x_k$ 的估计具有更高的对数似然贴合度，且标准误差更小。文中给出了递归估计的形式，即一种EM梯度粒子算法，该算法扩展了Lange等人关于最大似然平滑估计的工作。该算法具有显式的迭代更新步骤，而Ramadan等人提出的用于平滑的EM算法则缺乏此特性。本文发展了用于评估得分函数和观测信息矩阵的序贯蒙特卡洛方法，并推导了估计误差协方差矩阵的递归方程以计算标准误差。在线性系统情况下，该方法表明Rauch-Tung-Striebel平滑器是一种完全有效的平滑状态估计器，其协方差矩阵等于Cramér-Rao下界（即期望信息矩阵的逆矩阵）。此外，RTS平滑器与卡尔曼滤波器一致，且具有更小的协方差矩阵。数值研究验证了主要结果的准确性。